BOSTON — In recent weeks at Yankee Stadium, sprouting up through the thicket of pinstriped No. 2 jerseys and other remnants of the Derek Jeter era, there have been signs that Yankees fans are wrapping themselves in something other than nostalgia.

Now speckling the crowds are Gary Sanchez T-shirts and No. 24 jerseys, evidence, it would seem, that fans are investing emotionally and financially — those licensed jerseys retail for $129.99 — in the team’s slugging rookie catcher.

There is one T-shirt, though, that neatly captures what Sanchez has brought to the Yankees, who for more than a decade have been chasing yesterday, throwing money at players for what they have done, not what they might do.

It is a knockoff of the street artist Shepard Fairey’s iconic portrait of Barack Obama before the 2008 presidential election. Instead of Mr. Obama, it is Sanchez who gazes up confidently, above the block letters “HOPE.”

With the Yankees sliding out of playoff contention for another season — their 5-4 loss to the Boston Red Sox on Sunday night was their seventh defeat in eight games, leaving them four games out of a wild-card berth with 13 games remaining — this one at least will be ending with brighter prospects ahead.

When the team jettisoned Alex Rodriguez, Mark Teixeira announced his retirement, and several top veterans, including Carlos Beltran, were dealt to bolster the farm system by early August, it cleared the way for a number of prospects to audition.

There have been ups and downs, as might be expected of players finding their way, but Sanchez has been a revelation. With a third-inning home run Sunday night, he has hit 16 homers and driven in 30 runs in 41 games. The American League player of the month in August, he has also been dynamic defensively, rifling throws to all bases at just about any time.

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Sanchez after hitting a two-run homer Saturday on the road against the Red Sox.CreditMichael Dwyer/Associated Press

When Red Sox Manager John Farrell was asked before Saturday’s game if anything had struck him about the new-look Yankees, he said, “They have one hell of a catcher.”

The art of team building has changed drastically since the Yankees won their last title in 2009, having written big checks for Mark Teixeira and C. C. Sabathia, the best free-agent hitter and pitcher on the market. Now, as a result of the effects of revenue sharing, teams playing in the World Series are likely to ride homegrown talent there, as the Mets and the Kansas City Royals did last season.

Among this season’s top contenders, the Chicago Cubs, the Los Angeles Dodgers and the Red Sox are built around young cores that have blossomed years before hitting free agency. The Red Sox’ three young All-Stars — shortstop Xander Bogaerts and outfielders Jackie Bradley Jr. and Mookie Betts — will be paid less than $2 million combined this season and will not reach free agency until at least 2020, allowing the team to swing big (and sometimes miss) in free agency.

“Young, athletic, controllable players — that’s the lifeblood of any organization,” Farrell said. “And you see it all around the game.”

Sanchez, 23, did not necessarily sneak up on anyone. The Yankees signed him as a 16-year-old from the Dominican Republic for $3 million, the most they have bestowed on an amateur international free agent. But the Yankees do not have a good track record of patience — or development — with prospects.

A year ago at the trade deadline, General Manager Brian Cashman told teams that pitcher Luis Severino, first baseman Greg Bird and outfielder Aaron Judge were off limits. But he listened to offers for Sanchez.

“I’m glad for my sake that I didn’t do it,” Cashman said. “All the people guiding me through the process were saying: ‘This guy’s going to get there. He’s going to be a difference maker. He’s going to be special.’ We’ve turned down some pretty interesting packages over the years waiting for this to get here.”

Cashman added, “Most of the time it doesn’t work out this way, but in this case, he’s justifying the patience.”

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Sanchez merchandise has been selling well.CreditAlex Goodlett for The New York Times

The infusion of youth gave the Yankees a jolt that unexpectedly carried a disappointing, listless team back into playoff contention by early September. The clubhouse music became more lively when it was turned over to Judge, and the veterans say they have been reinvigorated by the rookies’ enthusiasm and naïveté.

Yet Sanchez, who has had the greatest impact, carries himself like an old hand. Naturally reserved and quietly confident, he gives interviews (conducted mostly through an interpreter) that reveal as little as his businesslike home run trots: head down and carried out with an absence of ebullience.

Sanchez has lived with his wife, Sahaira, and their 2-year-old daughter, Sarah, at his in-laws’ home in Yonkers since he was recalled. Sanchez’s grandmother, aunt and brother also live in New York, he said.

“Family is really important to him,” said reliever Nick Goody, who has frequently roomed with Sanchez on the road in the minor leagues since 2012.

The veteran’s mien that Sanchez displays does not desert him in the batter’s box. When he faced Tampa Bay’s Chris Archer, who is second in the American League in strikeouts, Sanchez fell behind by 1-2 in his first at-bat. He then laid off three tantalizing sliders just off the edge of the plate and drew a walk.

Although Sanchez later struck out and hit a home run off sliders, that initial at-bat stuck with Archer.

“Young hitters with the type of power he has are usually more aggressive; they’re more likely to chase,” Archer said. “So it showed me that you’ve got to prove you can throw it in the strike zone before he’ll start swinging at it. He laid off some good pitches. It was a great at-bat.”

Just how the Yankees will proceed in their rebuilding remains uncertain. Will Judge, the behemoth right fielder who has shown prodigious power and an alarming proclivity to strike out, develop into another potential cornerstone? Will Bird, who showed promise late last season, be the same player when he returns next spring from shoulder surgery?

And with a barren free-agent landscape for everything but relievers, will the Yankees package some of their other promising young talent for a pitcher who could join Masahiro Tanaka at the front of the rotation? Candidates include Jose Fernandez, Chris Sale and Archer, all of whom will probably be available, if at a steep price.

One area that is of less concern to the Yankees is catcher. Asked how he would assess Sanchez’s place next season, Manager Joe Girardi expressed little interest in peeking into the future, but he said the value the Yankees place on Sanchez was already evident.

“We’re showing you how much faith we have in him by hitting him third in the lineup and giving him a lot of responsibility,” Girardi said. “He’s catching a lot. He’s D.H.-ing. And that will continue because he’s playing at an extremely high level.

“We think he’s a really good player, and he’s shown that so far. But as far as worrying about where his place is next year, I’ve got four months or three months or whatever to worry about that.”

Girardi was speaking Friday, before the Yankees moved two losses closer to next season, although not through any fault of Sanchez’s. He drove in two runs with a double high off the Green Monster in Friday night’s loss, and he blasted a two-run homer off the Red Sox ace David Price that carried over the wall and onto Landsdowne Street in Saturday afternoon’s defeat. Even the two double plays Sanchez grounded into on Friday night were crackling hits that found the third baseman’s glove.

In fact, it was something like those double plays and the way they stood out from the rest that caught the eye of Archer. He watched more than a week ago as Sanchez picked Corey Dickerson off second base — from his knees. Archer also saw Sanchez nearly hit a home run off a pitch that was meant to be for an intentional walk, something Archer imagined that most hitters would not even be thinking about trying.

“These are things younger players do and the risks they’re willing to take,” Archer said. “You make a mistake? Who cares? Just play. Man, it’s fun to watch.”

It is a sentiment fans are beginning to share while watching the Yankees finish out this season, carried along by Sanchez and, whether they own a particular T-shirt or not, the audacity of hope.

A version of this article appears in print on , on Page D1 of the New York edition with the headline: The Silver Lining, Aglow in Pinstripes. Order Reprints | Today’s Paper | Subscribe